Among the difficulties of conventional autoencoders are the lack of uncertainty quantification in the estimation of model parameters and the need to tune regularization hyper-parameters. Bayesian neural networks address these by estimating the parameters of the model (weights and biases) as random variables, in contrast to single point estimates by backpropagtion learning that employ gradient methods [1]}, [2]}. The progress in neural networks has led to revolution in deep learning models; however, progress in Bayesian neural networks and Bayesian deep learning has been relatively slow due to certain challenges [3]}, [4]}. Bayesian inference implemented with canonical MCMC methods handles the large numbers of parameters involved with difficulty. However, variational inference provides an alternative Bayesian inference approach, with variational autoencoders [5]}, variational autoencoders and generative adversarial networks (GANs) [6]}, variational CNNs [7]}, [8]}, [9]}, [10]}, variational recurrent neural networks via long short-term memory (LSTM) networks [11]}, and variational graph neural networks [12]}, [13]}, [14]}. Further details about their applications are given in [3]}. We note that variational autoencoders also use a probabilistic representation of model parameters, as opposed to a single value representation in conventional autoencoders, facilitating uncertainty quantification [16]}, [17]}, [18]}.
